On non-factoriality of a class of simple hypersurface singularities

Question

Let $k$ be an algebraically closed field of characteristic $0$.

For which values of $n\ge 4$ the local ring $$R_n=k[[x,y,z,w]]/(x^2y+y^{n-1}+z^2+w^2)$$ is not a UFD ?

I know that any such ring is an integral domain in general, but I don't know any proof of non factoriality.

Please help.

Answer 1

I don't know how to show $R$ is an integral domain, but it is not a UFD.

For instance, $P = (y, z+iw)R$ is a height one prime ideal, and it is not principal (use Nakayama's lemma to check this).